I would not suggest changing this guitar from .009s to .012s without taking it to a guitar tech. The nut slots need to be sized to suit the gauges of strings on the guitar. Consider that the high E string on the new set would be .003 larger than the old one, and the low E string would be .008 larger than the old one; that is, the diameter of the new low E string is larger than the old one by almost the width of a light high E string. If the slots in the nut are too tight, the strings will bind there, and the guitar will not stay in tune as well. A guitar tech can widen the nut slots using a nut file for the new strings. If you wanted to go back to .009s, you could just change back, but it would be best to take the guitar back to a tech who can fill the nut slots and recut the nut for the lighter gauge strings.
Since the tension across the neck will change with the heavier strings, the truss rod will probably need to be adjusted to set the neck relief, and then the intonation will need to be set again. All this to say, going from .009s to .012s requires a setup, and a good guitar tech or luthier can do this for you.
If you just want to try out some flatwounds, TI makes flatwound .010s; you can probably get away with putting on some .010s without taking the guitar to a tech.